Level-rank duality of D-branes on the SU(N) group manifold

Abstract

The consequences of level-rank duality for untwisted D-branes on an SU(N ) group manifold are explored. Relations are found between the charges of D-branes (which are classified by twisted K-theory) belonging to su(N ) K and su(K) N WZW theories, in the case of odd N + K. An isomorphism between the charge algebras is also demonstrated in this case.

Level-rank duality of untwisted and twisted D-branes of the WZW model

“…In both the su(N) K and sp(n) k WZW theories, the charges of level-rank-dual untwisted D-branes are equal (modulo sign) [5,6], with a slightly more complicated relationship holding between the charges of twisted D-branes [7]. In the case of so(N) K , however, the charges of the D-branes do not exhibit any simple relationship under level-rank duality.…”

“…5 Integrable representations of an affine Lie algebraĝ K have nonnegative Dynkin indices (a 0 , a 1 , · · · , a r ) that satisfy r i=0 m i a i = K, where m i are the dual Coxeter labels of the Dynkin diagram forĝ K , and r + 1 is the rank ofĝ K .…”

“…More recently, it has been shown [5,6,7] that the untwisted and twisted D-branes in the boundary su(N) K WZW model [8]- [25] respect level-rank duality; that is, there exists a one-to-one map between the (equivalence classes of) D-branes of su(N) K and those of su(K) N . The open-string spectra associated with level-rank-dual D-branes are isomorphic, and the charges of level-rank-dual untwisted D-branes are equal (modulo sign), with a slightly more complicated relationship holding between the charges of twisted D-branes.…”

Twisted D-branes of the WZW model and level-rank duality

“…It has recently been shown [4,5] that levelrank duality also applies to untwisted and to certain twisted D-branes in the corresponding boundary WZW models [6]- [31]. (For a review of D-branes on group manifolds, see ref.…”

“…It was shown in refs. [4,5] that the charges of level-rank-dual untwisted D-branes of su(N) K and su(K) N are related bỹ Qλ = (−1) r(λ) Q λ mod x for N + K odd Q λ mod x for N + K even (except for N = K = 2 m ) (1.5) where r(λ) is the number of boxes in the Young tableau associated with the representation λ, whereλ is the level-rank-dual representation of su(K) N associated with the transposed tableau, and x = min{x N,K , x K,N }. For the remaining case, it was conjectured that on the basis of numerical evidence.…”